DOMAIN STRUCTURE TRANSITIONS OF METAL PLATELETS UNDER STRESS

HAL Id: jpa-00214509

https://hal.archives-ouvertes.fr/jpa-00214509

Submitted on 1 Jan 1971

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DOMAIN STRUCTURE TRANSITIONS OF METAL PLATELETS UNDER STRESS

J. Kaczer

To cite this version:

J. Kaczer. DOMAIN STRUCTURE TRANSITIONS OF METAL PLATELETS UNDER STRESS.

Journal de Physique Colloques, 1971, 32 (C1), pp.C1-251-C1-252. �10.1051/jphyscol:1971182�. �jpa-

00214509�

JOURNAL DE PHYSIQUE

Colloque C I , supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 251

DOMAIN STRUCTURE TRANSITIONS OF METAL PLATELETS UNDER STRESS

J. KACZER

Carnegie-Mellon University, Pittsburgh, Pa. 15213 (*)

R6sumB. - La structure de domaines d'un platelet monocristallin ferromagn6tique change de fac;on discontinue quand on applique une contrainte. A chaque transition correspond une certaine valeur propre de la contrainte. On derive A partir de l'knergie libre une expression qui donne le spectre de ces valeurs propres en fonction de la gbometrie et de l'6tat magnetique du platelet.

Abstract.

The domain structure of single crystal ferromagnetic metal platelets changes discontinuously on the application of stress. To each transition there corresponds a certain eigen value of the stress. From the free energy an expression is derived which gives the spectrum of these eigen value as a function of the geometry and magnetic state of the platelet.

In an earlier paper [I] we investigated the energy spectrum of the domain structure of cubic metal platelets under stress and found discrete eigenvalues of the stress o and the wall energy y(a) corresponding to an equilibrium domain configuration having n + ¹

1800-walls present in the platelet. This investigation gave rise to the question of the stability of each domain configuration and the transition between adjacent domain states as the stress is changed between two eigenvalues. Since the domain structure changes dis- continuously between states having say n and n + ¹

1800-walls, the problem is to determine the value of the stress corresponding to the condition of the transi- tion of state n to state n

1. The present paper deals with the conditions of domain structure transitions in metal platelets under stress, neglecting hysteresis effects.

The domain structure of the platelets of dimensions axbxc in their various states is shown in figure 1. The

FIG.

- Successive Equilibrium Domain Configurations of a Positive KT Parallelepiped with Applied Stress.

free energy of the platelet near local equilibrium is a function of the wall energy W which as was shown in [I] is a function of the stress a , the magnetoelastic stress energy in the triangular closure domains S, the demagnetising energy D caused by the displace- ment of the 900-walls from their local equilibrium posi- tions on application of the stress [2] and the state of the platelet n. For the total free energy per unit thickness of the platelet F wa can write

Here L = 3 ^{A,,, o} is the stress energy density,

(*)

IBM Postdoctoral Fellow on leave of absence from the Institue of Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia.

n + 1 the number of 1800-walls present, h/(n + ¹⁾

the height of the deformed triangular closure domains of basis a/(n + ^{1) and} B is a constant depending on the geometry of the platelet and on the number of clo- sure domains present.

While the meaning of W and S is clear we would like to discuss D in more detail. Application of the stress parallel to the 1800-walls decreases the volume of the closure domains in two possible ways. Either their height is reduced from a/2(n + 1) to h/(n + ¹⁾

as shown in figure 2a or their walls are deformed as shown in figure 2b. In the first case the platelet beco-

FIG.

TWO Possible Modes of Volume Decrease

Trian- gular Closure Domains due to Stress.

mes in the simplest case a quadrupole while in the second an octupole. Possibly both deformations are present simultaneously but for simplicity sake we shall only consider mode 2 a . As a result of this wall displacement alternating charges are set up on the 90°-walls which introduce a magnetostatic energy and therefore increases the free energy of the platelet near local equilibrium. The derivation of an exact expres- sion for the magnetostatic energy is very involved. We shall therefore derive an approximate expression. The surface density of this energy will be proportional to the square of the surface charge density o, and to the width of the strips a/2(n + 1). The constant of proportionality B will be in general a function of the geometry of the platelet and of the state of the platelet n.

1 2 a

E - - Ba,

- 2 2(n + 1)' For a; we get from figure 2a

which for small displacements (h z a/2) can be writ- ten

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971182

C I -

J. KAEZER

The total demagnetising energy for the platelet (per unit thickness) we get by substituting (3) into (2), multiplying by the width of the platelet a and doubling this (for both ends of the platelet).

Minimizing (1) with respect to h we obtain

After substituting (5) back into (I) we get for the free energy

F , = (n + ^{1) yb - ya} +

This equation differs from equation (7) in [1] by the last term which is a correction caused by the demagne- tising energy D. We now minimize with respect to n (see [I]) and obtain

which is a corrected form of eq. 8 in [I] and gives us the eigenvalues of the stress L: corresponding to equi- librium conditions. Substituting (7) back into (6) we get the equilibrium free energy F':. If we form the difference between F,, and F: we obtain a simplified equation

F = F , - ~ ~ = y b ( l - ~ j b ~ ~ ~ ) ( n + l - ~ ) ~ / ( n + l ) (8) where N2 is defined in eq. (7).

If we plot F as a function of L for constant n (i. e.

for a certain state of the platelet) we get parabolic dependencies as shown in figure 3.

The transition from state n - 1 to state n is a first order phase transition which takes place suddenly for a certain value of L lying between L:-~ and L,. To

solve for this value we equate the free energies Fn-, and F,, and obtain

Equation (9) defines the eigenvalues L: (and also of

d) at which transitions from state n - 1 to state n will take place not taking into account hysteresis effects. The actual instabilities of the domain structure will of course occur at values somewhat higher (for increasing L) and values somewhat lower than L:

(for decreasing L). To solve for the actual instabilities we would have to add to (1) another fourth order term in h which would make (1) non-linear. This term could only be obtained from anexact solution of the magneto- static energy calculation.

To arrive at an estimate of the constant B we com- pare our demagnetising energy with that of an infi- nite array of alternatingly charged strips of width a/2(n + 1) as originally calculated by Kittel [3].

Comparing this with eq. (2) we see that B will be of the order unity ( B will tend to 1,7 for n tending to infinity). Introducing this value into eq. (7) and (9) we find that for stresses commonly encountered both L/2 BM: and y / b ~ ~ : are much smaller than one.

Equations (7) and (9) therefore simplify to

(n + 1)' x a2 LE/2 byE (74 and

n(n + ^{1) x} a 2 LT/2 byT . ( 9 4 In these equations LE, yE andTLT, yT are the equili- brium and the transition eigenvalues of the stress and wall energies. Figure 4 shows the dependence of eq. (9a) in reduced stress p versus n for different values of the geometrical constant C.

1:-, L X L S ; , , L

FIG. 3.

Free Energy o f Platelet as a Function o f the Stress Energy L for Different States o f the Domain Structure.

FIG. 4.

Graph Giving the Eigenvalues

-

as a

Function o f C

4 b6o/a2 for n

..., 8. (60 is the

Thickness.)

References

[I] KACZER (J.), J. Appl. Phys., 1970, 41, 3336.

[2] GEMPERLE (R.), Phys. Stat. Solidi, 1966, 14, 121.

[3] KITTEL (Ch.), Rev. Mod. Phys., 1949, 21, 541.